A remark on a paper of B. R. Gelbaum

Summary In this paper the following generalization of a theorem by B.R. Gelbaum is proved:Let (K, d) be a compact, connected metric space. Let B denote the Borel sets of (K, d), and P be a probability measure on B with P(G)O for any nonempty open G, f, gC(K,d) independent random variables on (K,B,P) and let g satisfy the following assumption:There is an y _o such that g ^–1(y₀) is a finite nonempty set. Then f is a constant function.Examples show that the assumptions of this theorem are essential.